Finite groups of small orders

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This page lists, with small notes, finite groups of all small orders, starting from 1.

1

Further information: groups of order 1

  • Trivial group: This is the one and only one group with one element

2

Further information: groups of order 2

  • Cyclic group:Z2: This is the unique group of two elements. It is the cyclic group of order two. It's also the sign group and it's also the symmetric group on 2 elements.

3

Further information: groups of order 3

  • Cyclic group:Z3: This is the unique group of three elements. It is the cyclic group of order three. It's also the alternating group on three elements.

4

Further information: groups of order 4

  • Cyclic group:Z4: This is a cyclic group of four elements. It is the smallest nontrivial non-simple group.
  • Klein four-group: This is the direct product of the cyclic group of order 2, with itself. It also occurs as the group of double transpositions in the symmetric group of order 4. It is the smallest non-cyclic elementary Abelian group.

5

Further information: groups of order 5

6

Further information: groups of order 6

  • Cyclic group:Z6: This is the cyclic group on 6 elements. It is also the direct product of the cyclic group on 3 elements and the cyclic group on 2 elements.
  • Symmetric group:S3: This is the symmetric group on 3 elements. It is also the same as the dihedral group on 3 elements, and is the semidirect product of the cyclic group of order 3 with the cyclic group of order 2. It is also the holomorph of the cyclic group of order 3, or equivalently, the affine group of order 1 of the finite field of 3 elements. It is the smallest

7

Further information: groups of order 7

8

Further information: groups of order 8

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